Ollivier-Ricci Curvature of Riemannian Manifolds and Directed Graphs with Applications to Graph Neural Networks

TL;DR

This thesis explores Ollivier-Ricci curvature in metric spaces, connecting it to classical Ricci curvature, extending it to directed graphs, and applying it to network science and graph machine learning.

Contribution

It introduces novel extensions of Ollivier-Ricci curvature to directed graphs and demonstrates applications in network science and graph neural networks.

Findings

Connections established between Ollivier-Ricci and classical Ricci curvature.

Extended bounds and theorems to directed graphs.

Applications demonstrated in network science and graph neural networks.

Abstract

This thesis is an exposition of Ollivier-Ricci Curvature of metric spaces as introduced by Yann Ollivier, which is based upon the 1-Wasserstein Distance and optimal transport theory. We present some of the major results and proofs that connect Ollivier-Ricci curvature with classical Ricci curvature of Riemannian manifolds, including extensions of various theoretical bounds and theorems such as Bonnet-Myers and Levy-Gromov. Then we shift to results introduced by Lin-Lu-Yau on an extension of Ollivier-Ricci curvature on graphs, as well as the work of Jost-Liu on proving various combinatorial bounds for graph Ollivier-Ricci curvature. At the end of this thesis we present novel ideas and proofs regarding extensions of these results to directed graphs, and finally applications of graph-based Ollivier-Ricci curvature to various algorithms in network science and graph machine learning.